This channel problem dates back to the work of [1]}, who showed that when an inviscid bubble is injected into the channel filled with a viscous fluid, typically a single finger develops that propagates through the channel (see the numerical solution in Figure REF (e)). Since the work of Saffman and Taylor, extensive research has been carried out determining how the parameters of the model influence the width (relative to \(L\) ) and speed of this finger. In particular, for a fixed injection rate, as the surface tension parameter is increased, it is established that the width and speed of the finger increase and decrease, respectively. We refer the reader to [2]} and [3]} (and references therein) for a comprehensive overview of the problem. In this subsection (and subsection REF ), we restrict ourselves to the classic configuration where \(b\)  is constant, but we note that similar to the radial problem, our scheme can easily be used to study non-standard cases, such as those in the papers [4]}, [5]}, [6]}, [7]}.
